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Comment by norir | Hacker News Reader

It's interesting to me that I can't think of anyone remotely comparable to Euler in the public consciousness today. Even the work of someone like Erdos seems very esoteric by comparison and also was largely done in collaboration. Was Euler just born at the right time and picking all the low hanging fruit? Or maybe his immense creative production was a unique consequence of wealth plus limited distractions? I'm inclined to believe there are similarly talented individuals today but wonder if it is even possible to fully recognize them in the moment. Perhaps we will only discover the Eulers (and Shakespeares and Bachs) of today in a few hundred years time.

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Part of it is low-hanging fruit, certainly. Euler lived at a time when it was still possible to "know all math". That breadth of knowledge is simply not possible for a single human anymore; the discipline of mathematics is orders of magnitude larger. Comparable mathematicians today like Erdős and Terence Tao collaborate because they really can't learn the intricate details of every corner of math, so they collaborate with people who work in those corners instead. I think these are all people that have a deep understanding of the interconnected structures within math, and that makes them incredibly productive. I'm not going to try to compare the "level of genius" between Euler, Erdős, and Tao (although I think the latter two would readily claim Euler wins), but once you have a gift like that, there's a big difference between having all of mathematics in your head and not.

The low hanging fruit argument only takes you so far. How many other mathematicians in his epoch or before were able to pick as many low hanging fruits as him?

By all means he was a crazy outlier generational genius. A few others in history, like Archimedes and Newton, have been accused of "not leaving anything for anyone else to discover" as well. The question was: why do we not seem to see these crazy outliers anymore? The answer is certainly not that truly exceptional people simply stopped being born after the year 1800. The nature of what it could mean to "know everything" about a field has completely changed.

Also they lived in a time with far less education and communication.

Newton and Leibniz discovered calculus simultaneously. If calculus were a hot new idea now, dozens or hundreds of people would be discovering it simultaneously.

Look at NN/LLM AI for an example.

To be clear, I am not saying that Erdos and Tao are less talented than Euler. That seems impossible to say. But the magic of Euler is that his fingerprints are all over so many of the fundamental things that can be understood by a bright high school student but were mostly unknown before him. The work of figures like Erdos and Tao seems far, far less accessible in its present form at least and thus more limited in its overall impact.

Part of that, too, is the "founding father" effect. Most humans alive today are related to Genghis Khan and/or Charlemagne. It just takes time for new ideas to dissipate and cross-breed.

I find the notion of 'low-hanging fruit' in such contexts profoundly ahistorical. If graph theory was low-hanging why did it take thousands of years since Sumer or ancient Egypt? Or consider something from number theory: every other batch of students in a math camp I'm familiar with has someone who has 'proved' quadratic reciprocity for themselves -- and how could they not ? -- since childhood they have been immersed in a culture which points at it; while it took Euler roughly 40 years to even formulate the idea and then Gauss to prove it. It's not that - to use an anachronistic term - class field theoretic phenomena was not known to other cultures thousands of years ago.

(Btw there are many contemporary mathematicians at least at the level of Terence Tao but for some reason haven't been blessed by lay popularity -- mostly because Tao's math looks more school math like / familiar to non-math people than say Peter Scholze's)

It's low-hanging fruit because it depended on a bunch of mathematics and technology that Euler benefitted from: algebra, the printing press, and mass-produced paper. Sure, the Ancient Egyptians had papyrus but that is nothing compared to the volume of paper Euler had available to him.

Euler lived nearly three centuries after the invention of the printing press. He had vast numbers of books available to him and essentially unlimited paper to write on. He also had been tutored in algebra which remains the most important development in the history of mathematics. The abstract manipulation of symbols made possible by algebra is such an enormous leap over the geometric methods of the ancient mathematicians. It allows one to solve countless problems trivially in seconds which would take days to solve geometrically.

Your first sentence can be used for anything:

Etale cohomology was a low-hanging fruit because it depended on a bunch of mathematics and technology that Grothendieck benefitted from: algebra, the printing press, and modern transportation. Sure, Euler had horse drawn carriages but that is nothing compared to the speed of modern transportation that Grothendieck had available to him.

> If graph theory was low-hanging why did it take thousands of years since Sumer or ancient Egypt?

Because it wasn't low hanging thousands of years ago. It was only low hanging after an enormous body of foundational work was laid down over those thousands of years. And Euler knew all of it. It's no longer possible to know all of mathematics.

> every other batch of students in a math camp I'm familiar with has someone who has 'proved' quadratic reciprocity for themselves

This is exactly my point though: things get easier to understand over time as the more foundational mathematics gets laid out to prepare for them. Nowadays some of this stuff is considered basic. It's very "low hanging fruit" now, it's just that those summer-camp kids aren't making the discovery for the very first time. What point exactly are you defending here?

> Btw there are many contemporary mathematicians at least at the level of Terence Tao but for some reason haven't been blessed by lay popularity

I'm not sure why this needs to devolve into a contest. Terence Tao, Peter Scholze, whoever: they can't know all math anymore, like Euler did. That is ultimately why there are no more Eulers.

The difference is, to put in Rumsfeldian fashion, between known unknowns (kids proving QR now) and unknown unknowns (euler sensing QR, gauss proving it).

That Euler knew most of the math of his time is irrelevant. If one had any serious learning at any time in most of human history one would know all the math of their time.

Von Neumann is the closest comparison I can think of. A child prodigy who made a very large number of contributions to mathematics, computer science, physics, and engineering.

Reminds me of the early days of computer science, the 60s and 70s.

I once casually flipped through the CS journal papers archived in my university library, and those two decades were really fascinating.

1) All of the papers were simple, clear, and easy to follow. No greek symbols, no fancy maths, just straighforward pseudocode.

2) The algorithms being presented were new at the time, but were rather trivial, and could have been invented by any of us. They just hadn't been invented, formalised, and written down yet. It was basically a race starting from zero, and nobody had taken very many steps yet.

3) Every new algorithm was such a huge leap that it unlocked many new avenues of advancement for other algorithms. There was a period where it was a race of publishing these follow-up developments nearly as fast as people could type.

For a modern example of this, look at the pace of progress of generative AI such as LLMs and Diffusion Models. The first papers were almost trivial, but that "one clever trick" unlocked many more ideas and the rate of publication over the last twelve months has been insane. A lot of low-hanging fruit, a lot of road-blocks suddenly removed, etc...

It takes time for true quality to become apparent. Unfortunately the human who produced the work is long dead by then.

If humans ever started living much longer, we'd probably see a different attitude towards work.

Ramanujan died early. He is another one which comes to mind.

Riemann died young, too. I only really know him for the Riemann sum formulation of integrals (I have yet to learn complex analysis), but I wouldn't be surprised if he has some popular reach through the Riemann hypothesis.

Galois is yet another.

If only he had devoted some time to marksmanship

Round musket balls in hand packed smooth bore dueling pistols at 20 paces is inherently inaccurate - ten shots in a row from a pistol clamped in a vise will have a spread greater than the profile of an oppenent staning sideways.

Bullets with rifling came about circa 1820 (ish) but were not widespread by 1832 and traditional dueling pistols remained the norm for quite some time, the element of chance likely factored in as part of the hand of god influencing outcomes.

https://en.wikipedia.org/wiki/Duelling_pistol

    For some in the eighteenth century, duelling with less-accurate, smooth-bore weapons was preferred as they viewed it as allowing the judgement of God to take a role in deciding the outcome of the encounter.

There's not a lot of detail regarding the duel of Galois, his opponent isn't known for certain, nor the precise reasons, let alone the type of guns and ammunition used.

Another example: the notion of an n-dimensional geometric structure with intrinsic, variable curvature that serves as a mathematical foundation for the theory of general relativity originated with Riemann[1].

[1] https://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/WK...

Is Euler really well known in public? Einstein is way more well known than Euler, I'd guess. Even if you will ask most famous mathematician, it is unlikely that he is named. Probably Pythagoras.

I have no data to back it up.

Most people can't name any mathematicians, but amongst those that can I think he's top 5, behind more famous names like Newton, Archimedes and Euclid

And Gauss.

Sure, I didn't really mean the general public but meant scientific/mathematically oriented public. When I studied math in college, I found Euler everywhere to the point that it was hard not to stop and wonder how the hell this guy had so many fundamental discoveries. By contrast, I have awareness of present day figures like Tao and Erdos, but I don't really understand their work and perceive it to be specialized enough that it is unlikely to be widely understood in the way that say e^(pi * i) + 1 = 0 is.

This is purely an indictment of modern mathematics education. Anyone doing ordinary high school math should have had their ears full of Euler's contributions for at least a couple of years towards the end. Math teachers who don't mention Euler need to be put onto a mock Königsberg and forced to search for a solution.

For modern mathematics, I think Alexander Grothendieck was one of the most influencial talents in the last century. He unified large branches of mathematics in a very short period. He was more a bird than a frog [1], however.

[1] Bird vs. Frogs. https://www.ams.org/notices/200902/rtx090200212p.pdf

Stephen Wolfram's agent on line 2...

Maybe today's emphasis on "well rounded" education is a distraction from specialized talents.

Specializing is actually the "easier" path, and thus there isn't a lot of low hanging fruit anymore. There are more opportunities in the intersections of different fields (areas of mathematics).

Euler did his master’s dissertation on the philosophies of Descartes and Newton. He then joined the faculty at University of Basel in theology. I think this is ample evidence that he received a well-rounded education.

Not to mention his day job for a while as the chief cartographer in the Russian Empire.

Absolutely HARD disagree. Pythagoras, Aristotle, Pascal, Descartes, Newton, Franklin, DaVinci, (just to name a few) were all polymaths. They all left lasting impacts outside of a singular domain. It's because they were such well rounded individuals that they were so successful and capable of revolutionary discoveries and inventions. Our world is made up of models that are related with other models that are related based off of that relatedness. It's maddening. Being "well rounded" is vital.

Today that concept is watered down. A "well rounded" education is just taking a few classes that people hate and will blow off because to graduate they need to check some boxes so they can focus on doing one thing moderately well and finding their place as a cog in a machine that will abuse their ignorance. It's all mass produced conveyor belt education that manufactures young adults with little conventional wisdom. The more you lean into behaving like a part, the more you will be treated that way.

They didn't have to waste much time going to school -- and certainly not going to school with average people. That makes it a lot easier to get a good, well-rounded education -- if you are bright enough!

Euler was nothing if not diverse. Having a "liberal arts" / "great books" education from high school I have (anecdotal) first hand evidence in the impressive successes of my classmates in a WIDE variety of fields.

It's more that mathematics is solely consumed inside "mathematics for engineering 101" spread across N years.

Euler is in a class of one. Judging by the volume of work and the pleasure he takes in collaborations, Terence Tao seems to be having a positive impact. I am not a mathematician but I hear good things about the quality of his work.

> I am not a mathematician but I hear good things about the quality of his work.

Not to be rude, but saying this about a Fields medallist is somehow very funny.

I can see that. But, there are often controversies about who gets what award, even up to the Nobels. Since there are very highly qualified people posting here, I just wanted to be clear that I have no ability to understand Tao's work and am going by acclaim.

Edward Witten bestrides mathematical physics in an Euler-likes way. In math/physics circles, ask the smartest person you know who is the smartest person they know and sooner or later all roads lead to him.

If not "low-hanging fruit" at least "first mover advantage."

Karpathy! Carmack!

Carmack? I love him but that can’t be a serious suggestion

Yeah Carmack is no doubt a 20x engineer, but I wouldn’t put him in the same category of prolific genius as Euler. As far as prolific engineers go, I’d say Fabrice Bellard would be closer (but still not even close) to an Euler.

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